Curvature Sign Rigidity and Sharp Pointwise Pinching Thresholds
Abstract
We study connected Riemannian manifolds on which either the sectional curvature or the Ricci tensor is, at each point, strictly positive, strictly negative, or zero, and ask whether the two signs can coexist under pointwise pinching. A general support-rigidity theorem for positive semidefinite divergence-free symmetric tensors is the common analytic mechanism. For sectional curvature in dimension n≥ 3, any locally uniform positive lower bound for the absolute pointwise pinching ratio rules out a change of sign. With a fixed pinching constant δ0, the flat set has no C1 hypersurface piece; it is empty when δ0>1/2, and is locally porous when δ0=1/2. These conclusions are sharp in several senses: there are smooth conformally flat local metrics whose sectional curvature changes sign across a flat hypersurface when the pinching degenerates, and there are closed exactly 1/q-pinched metrics on spheres with isolated flat points. For Ricci curvature, the sharp threshold is δc=1/(n−1). A locally uniform gap δRic>δc forces one Ricci sign globally. Conversely, for every 0<δ<δc there are local sign-changing metrics with exact pointwise pinching δRic≡δ, and exact local examples also exist at δ=δc. Pointwise strictness δRic>δc is insufficient if the gap collapses at a Ricci-flat interface. For every subcritical δ we give explicit closed metrics on 𝕊1×𝕊n−1, and complete periodic lifts to ℝ×𝕊n−1, whose optimal global lower pinching constant is exactly δ. Finally, we prove two ansatz-specific obstructions to compactifying the exact local constructions. The main content of the proof is generated by ChatGPT 5.6 sol and verified by the authors.
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BibTeX
@misc{gao2026curvature,
title = {{Curvature Sign Rigidity and Sharp Pointwise Pinching Thresholds}},
author = {Minbo Gao and Yuhang Liu and Genyuan Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07252},
eprint = {2609.07252},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07252},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Curvature Sign Rigidity and Sharp Pointwise Pinching Thresholds
AU - Minbo Gao
AU - Yuhang Liu
AU - Genyuan Zhang
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07252
UR - https://arxiv.org/abs/2609.07252
AB - We study connected Riemannian manifolds on which either the sectional curvature or the Ricci tensor is, at each point, strictly positive, strictly negative, or zero, and ask whether the two signs can coexist under pointwise pinching. A general support-rigidity theorem for positive semidefinite divergence-free symmetric tensors is the common analytic mechanism. For sectional curvature in dimension n≥ 3, any locally uniform positive lower bound for the absolute pointwise pinching ratio rules out a change of sign. With a fixed pinching constant δ_0, the flat set has no C^1 hypersurface piece; it is empty when δ_0>1/2, and is locally porous when δ_0=1/2. These conclusions are sharp in several senses: there are smooth conformally flat local metrics whose sectional curvature changes sign across a flat hypersurface when the pinching degenerates, and there are closed exactly 1/q-pinched metrics on spheres with isolated flat points. For Ricci curvature, the sharp threshold is δ_c=1/(n−1). A locally uniform gap δ_Ric>δ_c forces one Ricci sign globally. Conversely, for every 0<δ<δ_c there are local sign-changing metrics with exact pointwise pinching δ_Ric≡δ, and exact local examples also exist at δ=δ_c. Pointwise strictness δ_Ric>δ_c is insufficient if the gap collapses at a Ricci-flat interface. For every subcritical δ we give explicit closed metrics on 𝕊^1×𝕊^n−1, and complete periodic lifts to ℝ×𝕊^n−1, whose optimal global lower pinching constant is exactly δ. Finally, we prove two ansatz-specific obstructions to compactifying the exact local constructions. The main content of the proof is generated by ChatGPT 5.6 sol and verified by the authors.
ER -